Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214×C4 |
| direct product | C23×#388 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C23 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C27 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
25810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 18 points:
(1,2);(3,4);(5,6);(8,10)(12,13)(15,16,18,17);(11,12,14,13)(15,17,18,16);(7,8)(9,10)(11,13,14,12)224 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#2669, C23×#2358, C23×#2424, … |
| maximal normal subgroup of | C23×#2669, C23×#2358, C23×#2424, … |
| maximal quotient of | C22×#13873, C22×#14696, C22×#14019, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532432);